Noise and Fluctuations: An Introduction

by D. K. C. MacDonald

Overview

An understanding of fluctuations and their role is both useful and fundamental to the study of physics. This concise study of random processes offers graduate students and research physicists a survey that encompasses both the relationship of Brownian Movement with statistical mechanics and the problem of irreversible processes. It outlines the basics of the physics involved, without the strictures of mathematical rigor.
The three-part treatment starts with a general survey of Brownian Movement, including electrical Brownian Movement and "shot-noise," Part two explores correlation, frequency spectrum, and distribution function, with particular focus on application to Brownian Movement. The final section examines noise in electric currents, including noise in vacuum tubes and a random rectangular current. Frequent footnotes amplify the text, along with an extensive selection of Appendixes.

Product Details

ISBN-13:	9780486174464
Publisher:	Dover Publications
Publication date:	01/23/2013
Series:	Dover Books on Physics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	128
File size:	4 MB

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NOISE AND FLUCTUATIONS

AN INTRODUCTION

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By D.K.C. MacDonald

Dover Publications, Inc.

Copyright © 2006 Dover Publications, Inc.
All rights reserved.
ISBN: 978-0-486-17446-4

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CHAPTER 1

General survey

1.1 INTRODUCTION

In this first Chapter, I would like to try to outline some of the main problems that arise from physical fluctuations or "noise," and to try to dispel some of the mystery that often seems to surround the significance of fluctuation phenomena. First, let us clear up the matter of terminology. Acoustically a regular tone or sequence of tones that may either convey obvious information in some sort of code, or simply present a musical melody, can usually be distinguished quite readily from what we call "noise." In the limit, audible noise is just a more or less random jumble of tones of a wide range of frequencies having no particular connection with one another, and such that if displayed on an oscillograph the noise would give rise to a more or less jagged and very irregular pattern whose amplitude varies randomly between fairly wide limits. This is in strong contrast to the pattern produced by a regular and unchanging single musical tone (see Figs. 1a and 1b).

This contrast between a regular pure tone and "noise" can also be seen in the degree of predictability. For a regular musical tone the pattern on an oscillograph and, of course, the corresponding audible sound are almost perfectly predictable, whereas with noise both the visual pattern and perhaps the audible content are largely unpredictable. It may be partly the latter aspect that makes audible noise sometimes so objectionable, although I am not suggesting that perfectly predictable experiences are necessarily the acme of human enjoyment.

If now we extend these ideas to other fields of physical behavior such as electrical oscillations, then the term "noise" can be applied (if one is not too squeamish) to any physical variable that is not behaving in an entirely regular and predictable manner. If we turn up the volume control of an ordinary broadcast receiver when no station is tuned in, then sooner or later a general "mush" or rasping "noise" will be heard from the loudspeaker. If there is no antenna connected, a radio engineer might then say that we were listening to the electrical noise in the resistors and tubes of the first stage. A more "respectable" physicist might prefer to say that the "spontaneous fluctuations" of electric charge were being amplified to produce a more or less random, audible sound. Or, if we listen to rain on a metal roof, we hear a general rattling "noise" caused by the impacts of the individual raindrops which are by and large random and unpredictable in detail; a physicist will point out that the time-average of the impulses gives rise to a steady pressure on the roof, whereas the fluctuations set up random oscillatory vibrations in the roof which in turn we hear as the "noise."

Why is the whole field of "noise" or fluctuation phenomena of considerable importance in physics? There is no single reason, and perhaps that is part of the fascination of the subject. Here are one or two examples. First, assume we are designing a radar set, or alternatively that we are trying to make a very sensitive radio receiver to pick up any radio emissions (in themselves a form of electrical "noise"!) that may be coming to the earth from outer space (e.g., the sun or some distant stellar object). With a radar set the useful range of the equipment will depend on how much power we send out from the transmitter, and also on the smallest reflected power from the distant object that we can detect with our receiver. From that point of view any improvement that we can make in the ability of our radar receiver to detect a weak signal is precisely equivalent to a corresponding power increase in our radar transmitter, and may therefore represent a very large economic saving. If we are engaged in radio-astronomy, an improvement in our receiver may enable us to observe with certainty more distant galactic objects. In either case the fundamental limit for the ability of our receiver to detect very small incoming signals will be set primarily by the magnitude of the "noise" or random fluctuations of electric charge in the input circuit and in the first amplifying stage of the receiver. Hence a theoretical understanding of the sources of such "noise" may enable us directly to design a better receiver. And this, of course, applies to many other fields where sensitive amplifiers and detectors may be required. In various branches of physics very sensitive measurements are often now being made to detect small effects predicted by developments of quantum theory. For example, today (mid-1961) experiments are reported on small superconducting cylinders to demonstrate that magnetic flux is quantized in units of hc/2e (~2 × 10-7 gauss cm2). Such experiments demand very careful attention to detail, and naturally care must be taken at many points to ensure that "noise" and random fluctuations in the apparatus are kept to an absolute minimum if we are to observe with any confidence such small effects.

Secondly, consider what we might observe with a sensitive radio receiver when the antenna is pointed at the sun for example. In general, if our antenna is sufficiently sensitive (and can discriminate within a sufficiently small solid angle), we shall notice increased "noise" at the output of our receiver arising primarily from incoherent electromagnetic radiation of one kind and another from the sun and covering a wide frequency spectrum. If we have a basic theory of how hot bodies of one kind and another emit such "noise radiation," we may then be able to use our observations of "noise" to obtain some direct information about the properties of the sun. A rather related sort of application is to use directly the electrical noise, or spontaneous fluctuation of some other physical quantity in a body, to measure the temperature at which that body finds itself. When the noise from a passive electrical resistor is used in this way to indicate the absolute temperature, the set-up is usually known as a "noise thermometer."

Thirdly, let us consider a dissipative process involving a parameter such as viscosity. Consider a small sphere falling steadily through a bath of oil under the force of gravity. The viscous force retarding the motion of the sphere is, of course, determined by some average of the vast number of rapidly varying individual forces exerted by the surrounding oil molecules. At the same time, however, precisely because the average (and irreversible) viscous friction does arise from a large number of individual and more or less independent components, this viscous force must also have a fluctuating component. Einstein (1905, 1906) first showed that there was an essential and fundamental connection between the average force (viscosity or mobility) and the fluctuating component (giving rise to the so-called Brownian Movement of the particle); moreover it may be quite possible under suitable conditions to observe directly this fluctuating component. There are in fact situations where it might be simpler to observe the Brownian Movement and hence, using Einstein's analysis, to deduce the viscosity; alternatively a direct measurement of viscosity may enable us to predict the diffusion of a typical atom, which in effect is just Brownian Movement. Apart from this it may also be that theoretically we can estimate rather directly the behavior of the fluctuating force contributions arising from individual atoms in the environment, and hence in turn be able to predict the average, irreversible viscous force that will arise.

Lastly, there are often situations (and not necessarily confined to physics) where some knowledge of the analysis of fluctuations (or perhaps more simply of statistics) may greatly simplify the solution of a problem. It might be argued that when traffic begins to emerge from the center of a large city at the end of the working day, the behavior of an individual motorist could be regarded as almost random! This is one of the paradoxes of the situation; any human being is probably insulted if one suggests that his progress through life resembles in any way a "random walk," but it is not hard to convince him that this is a good description of other people's behavior, particularly when they get in his way! Now Einstein first showed us that diffusion on the macroscopic scale can be interpreted as the "random walk" of a large number of individual particles or molecules on the microscopic scale, so here we might try reversing the situation. In order to make a first attempt, perhaps as a traffic engineer, to deal with the outflow of vehicles we might look at the solution of the classical diffusion equation under appropriate boundary conditions. The point is here that it is far more convenient to treat this problem as if each individual car were moving "randomly" than if we were to attempt the more or less hopeless task of really trying to predict individually the motion of the many thousands of vehicles. And very frequently we may "get away with it"—at least to a first approximation—because, viewed on the grand scale, the individual variations (or fluctuations!) in the routes and goals of the various cars will be reasonably well approximated by assuming that they are random.

At the same time this approach points out possible limitations in the statistical attack on a problem. If we treat our traffic problem as a purely statistical "diffusion" situation, then this might well enable us to design roads capable of handling the "expected" (i.e., average) outflow of traffic without difficulty. On the other hand, precisely because we really are dealing with individual human beings driving individual vehicles, there may still be a great deal of individual frustration involved because the actual behavior of an individual car may fluctuate—and we emphasize the word—from the statistical average. This in turn points a way toward improving our approach to this particular problem if we are not merely a traffic controller, but a traffic controller who is interested in minimizing frustration to individual human beings. Namely, we work out first the average solution, and then turn our attention to the expected fluctuations from that average; if we then make a reasonable allowance for these fluctuations from the average behavior, we shall at least be making some reasonable provision for the individual.

A rather poignant case of this type could arise if we were trying to provide for human beings with very low incomes. A politician might be advised that the average income required to provide subsistence for an average man was fifteen dollars per week, and hence believe sincerely that provision of this amount for each would mean subsistence for all.

However, a little thought might suggest that because of the inevitable fluctuations of individual human beings from the average, about half the people concerned might actually be going more or less hungry (see Fig. 2)!

Very broadly speaking, I would say that the two following aspects of fluctuation theory are of primary importance.

1. If we are dealing with a large collection of particles, atoms, or some other sort of entities, then for many purposes a straightforward and immediate average of the behavior (e.g., average pressure due to raindrops on the roof, average number of people leaving football matches, etc.) may do. But if we look closer it is often the fluctuations that tell us that we are, in fact, dealing ultimately with individual entities, and so from one point of view we can say that fluctuations may show the intrusion of "mechanical" concepts into an otherwise purely statistical argument.

2. As things stand today the almost "orthodox" approach in physics to any problem involving large assemblies of individual entities is to appeal to Statistical Mechanics. Broadly speaking, statistical mechanics will tell us about the average behavior of a system in thermal equilibrium with its surroundings, but statistical mechanics by itself is rather reluctant to say anything about how long it takes for this equilibrium condition to be reached, or in what manner this equilibrium is approached. I would argue that this is a field where fluctuation theory is likely to tell us something of value.

Actually, statistical mechanics can often tell us not only a lot about the average state of a physical system but also make predictions about the over-all magnitude of the average, or "expected," fluctuations from this average state. But what it does not tell us of itself is anything about the rapidity or time-variation of these fluctuations, which can be vitally important because a fluctuation that happens very rapidly may be of no significance in some physical situations, and thus a knowledge of only the over-all magnitude of the fluctuations could be quite misleading. Thus, broadly speaking, we might suggest that a more respectable name for noise and fluctuations theory could perhaps be "time-dependent statistical mechanics."

Let us bear in mind, however, that rather generally, both in this book and elsewhere in this field, we are directly concerned with analyzing the fluctuations, or "noise," of macroscopically observable quantities (the motion of a Brownian particle, the fluctuations of electric charge or current in some circuit, or even the fluctuations of the incomes of human beings!). This then generally means that each "segment" or "element" comprising the observed noise or fluctuation is distinguishable in time or in space from every other segment. Our "counting," or statistical analysis, can proceed quite straightforwardly on a so-called classical basis. It is only if we are directly concerned with the detailed statistical mechanics of assemblies of so-called fundamental particles or quasi-particles (such as electrons or photons) that we must exercise particular care in our "counting" procedure (leading more specifically to Fermi-Dirac or Bose-Einstein statistics; see Comment to Appendix I, the remarks on p. 64, and note (3) on p. 69).

1.2 BROWNIAN MOVEMENT

1.2.1 Historical Origins

Let us now follow through in broad outline the history of the Brownian Movement and its analysis, together with the more recent understanding of the fundamentals of electrical fluctuations (electrical Brownian Movement, i.e., the so-called Johnson noise or thermal fluctuations; and "shot-noise"). Thereafter in subsequent chapters we shall try to discuss in a little more detail some particular aspects of fluctuations analysis (or, if one prefers, "noise" problems).

The term Brownian Movement derives from the biologist Robert Brown (1828), who had been observing the tiny pollen grains of a plant under his microscope. In his words: "While examining the form of these particles immersed in water, I observed many of them very evidently in motion. These motions were such as to satisfy me ... that they arose neither from currents in the fluid, nor from its gradual evaporation, but belonged to the particle itself." Brown examined particles from very many substances (even including a fragment from the Sphinx!), and found that this motion persisted for every kind of substance examined. Today the obvious conclusion is that the source of the phenomenon must, of course, be the irregular bombardment of the particles by the molecules of the surrounding fluid, but Brown himself drew the conclusion at first that, because of the universality of the effect, he had encountered the existence of some elementary form of life in all organic and inorganic matter! During the subsequent seventy years or so until Einstein's (1905, etc.) remarkable analysis, many other experiments were made and there was much theoretical speculation about the effect. Proposals that the fundamental origin lay in some kind of electrical force (Jevons, 1869), evaporation of the liquid (Wiener, 1863; Gouy, 1889), or mechanical shocks (Exner, 1900), were disposed of in turn; the Brownian Movement persisted unchanged after the sample had been kept in the dark for a week (Meade Bache, 1894) or after an hour's heating (Maltezos, 1894), and it became clear ultimately that the effect must be quite fundamental.

1.2.2 (Erroneous) Arguments against Molecular Origin

It was Einstein (1905, etc.) who in a series of classical papers was the first to provide a clear theoretical analysis showing that the Brownian Movement arose directly from the incessant, and more or less random, bombardment of the particles by the molecules of the surrounding liquid. Before that von Nägeli (1879) had already suggested the possibility that molecular bombardment was the cause of the movement, but he discarded this proposal on the following grounds.

(Continues...)

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Table of Contents

1. General survey
2. Correlation, frequency spectrum, and distribution function
3. Noise in electric currents
Appendixes
References
Index